On the Fundamental Group of a Variety with Quotient Singularities

Abstract

Let k be a field, and let |$\pi :\tilde {X} \longrightarrow X$| be a proper birational morphism of irreducible k-varieties, where |$\tilde {X}$| is smooth and X has at worst quotient singularities. When the characteristic of k is zero, a theorem of Kollár [7] says that π induces an isomorphism of étale fundamental groups. We give a proof of this result which works for all characteristics. As an application, we prove that for a smooth projective irreducible surface X over an algebraically closed field k, the étale fundamental group of the Hilbert scheme of n points of X, where n>1, is canonically isomorphic to the abelianization of the étale fundamental group of X. Kollár has pointed out how the proof of the first result can be extended to cover the case of quotients by finite group schemes.